Chapter 1
Reification of Boolean Logic
Exercises
1.1 (a) Design a feedforward network to divide the black dots from other corners with
fewest neurons and layers. Please specify the values of weights and thresholds.
001
011
101
111
000
010
100
110
(b) Is it possible to do (a) with a single neuron? Why?
1.2 Consider the neural network in Fig. P1. The value of each neuron can be 0 or 1
and the activation function used is f (net) = {1, net>0; 0, net<0}. Each neuron
decides its own value according to the values of neighbors. We can adjust the
weights of neurons so that every neuron will have distinct set of action rules (e.g.
{000|0, 001|1, …, 111|0} is a set of action rules). If two neurons that have
different sets of action rules are considered to be different kinds, then, how many
kinds of neurons can we have in the network?
Fig. P1.
1.3 Design a feedforward network which provides the classification of the following
pattern vectors:
Class 1:
X = [0 0 0 0 ]t, [0 0 1 0 ]t, [0 0 1 1 ]t, [0 1 0 0 ]t, [0 1 0 1 ]t, [0 1 1 0 ]t, [0 1 1 1 ]t, [1
0 1 0 ]t, [1 1 0 0 ]t, [1 1 1 0 ]t, [1 1 1 1 ]t
Class 2:
X = [0 0 0 1 ]t, [1 0 0 0 ]t, [1 0 0 1 ]t, [1 0 1 1 ]t, [1 1 0 1 ]t
Please specify the value of weights and thresholds and use as few neurons and
layers as possible.
1.4 Please refer to p.217 in “Introduction to Artificial Neural Systems” by Jacek M.
Zurada.
M(J, n) = 2J, n ≧ J.
n
Is it enough to use J neurons as hidden layer to represent all 2 2 Boolean
functions when n = J?
1.5 To compare neural model equation y = σ(Σ wixi + w0) with Boolean equations
which use only operators ∩, ∪ and ~.
(1) List all different Boolean equations in the forms as follows.
(a) Y = a1X1 ∪ a2X2 ∪ a3X3.
(b) Y = a1X1 ∩ a2X2 ∩ a3X3.
(c) Y = a1X1 ∪ a2X2 ∩ a3X3.
(d) Y = a1X1 ∩ a2X2 ∪ a3X3.
(a1, a2, a3 can be ~ or empty string.)
And specify which equations appear in neural model’s 104 functions, which
didn’t.
(Hint: You can use the 3-cube to represent Boolean equations)
(2) We have five methods to cut the cube to get the 104 functions. For each
cutting method, write a corresponding Boolean equation in the forms given
above.
1.6 We have these training patterns:
0: (0, 0, 0), (0, 1, 1), (1, 1, 0)
1: (0, 1, 0), (1, 0, 0), (1, 1, 1).
The training result:
- 0.9 1.5 - 1.0 
W1  2.5 - 3.7 0.37
2.5 - 2.0 4.0 
W2  5 6 6
Bias 1  - 0.95 - 2.0 - 3.1
bias
bias
Bias2 = -3
(1) Write the Boolean function of training patterns. (Make simplification)
(2) Write the nested Boolean function of the network architecture for each
neuron. Notice that the Boolean function you wrote must be in the forms
given in problem 1.
(3) Prove that function (1) and (2) are equivalent.
(4) (0, 0, 1) and (1, 0, 1) didn’t appear in training patterns. So, we can get four
different Boolean functions. Please choose the simplest Boolean function and
compare it with the output of the above neural network of these two patterns.
1.7 Training patterns:
(1) Write the Boolean function.
(You must make simplification)
(2) Design a 3-2-1 neural network
to classify these patterns. If it
cannot classify these patterns
correctly, construct your own
neural network to classify these
patterns.
1.8 Boolean functions and neural network have a one to one map in architecture.
Please find the map and make a description of it.
Hint: y  ( x1  x2  x3 )  ( x1  x2  x3 ) can map to this neural network
y
2
1
-1
2
1
x1
1
-1
x2
1
3
1
1
x3
1.9 Find the coloring graph solution for x1  x2  x3  Y .
Remember x1  x2  x3  Y is,
coloring graph is similar to that in page 79
“Turing Machines” by J. E. Hopcroft, 70-80.
1.10 Proposition: The Boolean function is defined as: there are six Boolean variables
as input (x1, x2, …, x6) of this function, the output of this function is 1 only when
any two variables are 1, (more than two variables or less than two variables are 1,
the output of this function is 0.)
(a) Use Boolean algebra or Boolean function to express this proposition.
(b)
(c)
(d)
Write a program (Turing machine, Lisp, C, or other programs) to
simulate this expression, the input of the program is these six Boolean
variables, the output of the program is according to the proposition.
Discuss the trained weights of the multilayer network in homework
for the above proposition. Can you figure out the meaning of those
weights?
Construct a multilayer network for this proposition. Use six neurons
in input layer and one neuron in output layer.
1.11 In class we discussed a neuron with two Boolean variables as input. This neuron
can simulate 14 Boolean functions (14 Boolean states) excluding the XOR.
Assume the neuron is in state Si, which is one of the 14 Boolean functions. When
we slightly tune the weights w1 w2 w3 of this neuron the current state Si will
change to Sj first. Discuss all possible such first Sj when this neuron is in Si for
all 14 states, i = 1~14.
1.12 Write an algorithm to train the neural network in problem 2 by its training
patterns. Notice that the algorithm must be constructed by the following
subroutine:
train_one_neuron( x, d, w, y, Δw)
input: x, d, w. output: y, Δw.
x: an array of the input value.
d: a desire output.
w: the weights.
y: the output.
Δw: an array of the value of the weights should be added.
1.13 In appendix A we showed a modular design method for training ANN on data
flow machine. Each neuron’s desire response
is inducted from choosing bipolar sigmoid function  (u ) 
2
 1 . Please try to
1  e u
formalize the desire response when we choose unipolar sigmoid function
 (u ) 
1
1  e u
1.14 Discuss and analyze the results obtained in training a 1-3-2 ANN. You can use
the following results or your own results. The input is a continues number between 0
to 1 and output is a ‘S’ curve in the 2D plane.
001
011
12
111
1 101
10
0.8
011
110
8
111
0.6
6
010
000
0.4
101
4
0.2
0 100
0
010
0
0.2
0.5
0.4
0.6
2
001
100
0
000
0.8
110
1
1
-2
-10
-8
-6
-4
-2
0
2
1.15 (a) A dataflow information processing architecture is a MIMD architecture
without global or shared memory in which each processing element only operates
when all of the necessary information that it needs to function has arrive. Show that
neural networks are dataflow architectures.
(b) Invent a concrete, detailed example of a dataflow architecture that is not a
neural network.
1.16 We have learned basic type of neural network:
y  f (i1 wi xi   )
n
We can directly use it to simulate Boolean logic. For example: OR Boolean
function: y=x1 OR x2, where x1, x2, y are Boolean variables having values {0,1}.
When above network use ‘hardlim’ active function, its output y=x1 OR x2.
Another example is AND Boolean function: y=x1 AND ¬x2
Please check the truth table for it by yourself and answer questions below:
(a) draw the neural network that perform:

y  x  x  x  x
1
2
3
4

Note: the weights and bias should be {-1,1} for simplicity.
(b) draw the neural network that peform:

y  x  x  x  x
1
2
3
4

Note: the weights and bias should be {-1,1} for simplicity.
(a) Can you formalize how to set up the weights and biases for a given OR
expression?
(b) Can you formalize how to set up the weights and biases for a given AND
expression?
(c) Try to use a 3-layers neural network for simulating the DNF equation:
y   x  x  x    x  x  x 
2
3
4
5
 1
 3
(f) The neural network’s outputs can be feedback as inputs like:
It can simulate many kinds of dynamic process like gene regulation and etc. If
n Boolean variables x1, x2, … , xn are changed by time and its value are known
for a period of time. We can build ANN model like above for these n-variables.
Please try to build an ANN model for the given 4-variables.
Time
x1
x2
x3
x4
1
1
0
1
1
2
0
1
0
0
3
0
0
0
1
4
1
1
1
1
5
0
1
1
0
6
1
1
0
0
7
1
0
0
0
8
0
1
0
1
(Hint: first rewrite x1= F(x1, x2, x3, x4) and get its DNF)